Topological classification of certain nonorientable 4-manifolds with cyclic fundamental group of order 2 mod 4

TL;DR

This paper extends the classification of nonorientable 4-manifolds with cyclic fundamental groups from order 2 to order 2p for odd p, using a cut-and-paste approach and existing surgery theory results.

Contribution

It introduces a method to classify a broad class of nonorientable 4-manifolds with cyclic fundamental groups of order 2p, expanding previous classifications.

Findings

Classification extends to groups of order 2p for odd p

Uses cut-and-paste construction and existing surgery results

Presents open questions for complete classification

Abstract

We show that the classification up to homeomorphism of closed topological nonorientable 4-manifolds with fundamental group of order 2 due to Hambleton-Kreck-Teichner can be used to classify a large set of such 4-manifolds with cyclic fundamental group of order 2p for every odd $p > 1$. This is done through a simple cut-and-paste construction, and classical and modified surgery theory are used only through results of Hambleton-Kreck-Teichner and Khan. It is plausible that this set comprises all closed topological nonorientable 4-manifolds with $\pi_1 = \Z/2p$. We collect several interesting questions whose answers would guarantee a complete classification.